Fourier expansion in variational quantum algorithms

The Fourier expansion of the loss function in variational quantum algorithms (VQA) contains a wealth of information, yet is generally hard to access. We focus on the class of variational circuits, where constant gates are Clifford gates and parameterized gates are generated by Pauli operators, which covers most practical cases while allowing much control thanks to the properties of stabilizer circuits. We give a classical algorithm that, for an $N$-qubit circuit and a single Pauli observable, computes coefficients of all trigonometric monomials up to a degree $m$ in time bounded by $\mathcal{O}(N2^m)$. Using the general structure and implementation of the algorithm we reveal several novel aspects of Fourier expansions in Clifford+Pauli VQA such as (i) reformulating the problem of computing the Fourier series as an instance of multivariate boolean quadratic system (ii) showing that the approximation given by a truncated Fourier expansion can be quantified by the $L^2$ norm and evaluated dynamically (iii) tendency of Fourier series to be rather sparse and Fourier coefficients to cluster together (iv) possibility to compute the full Fourier series for circuits of non-trivial sizes, featuring tens to hundreds of qubits and parametric gates.Comment: 10+5 pages, code available at https://github.com/idnm/FourierVQA, comments welcom

Conformal symmetry in quasi-free Markovian open quantum systems

Conformal symmetry governs the behavior of closed systems near second-order phase transitions, and is expected to emerge in open systems going through dissipative phase transitions. We propose a framework allowing for a manifest description of conformal symmetry in open Markovian systems. The key difference from the closed case is that both conformal algebra and the algebra of local fields are realized on the space of superoperators. We illustrate the framework by a series of examples featuring systems with quadratic Hamiltonians and linear jump operators, where the Liouvillian dynamics can be efficiently analyzed using the formalism of third quantization. We expect that our framework can be extended to interacting systems using an appropriate generalization of the conformal bootstrap.Comment: 15 pages, supplementary Wolfram Mathematica notebook available at https://github.com/idnm/third_quantization v2: minor revision (references added, typos corrected) v2: Minor revisions done and typos correcte

Demonstration of a parity-time symmetry breaking phase transition using superconducting and trapped-ion qutrits

Scalable quantum computers hold the promise to solve hard computational problems, such as prime factorization, combinatorial optimization, simulation of many-body physics, and quantum chemistry. While being key to understanding many real-world phenomena, simulation of non-conservative quantum dynamics presents a challenge for unitary quantum computation. In this work, we focus on simulating non-unitary parity-time symmetric systems, which exhibit a distinctive symmetry-breaking phase transition as well as other unique features that have no counterpart in closed systems. We show that a qutrit, a three-level quantum system, is capable of realizing this non-equilibrium phase transition. By using two physical platforms - an array of trapped ions and a superconducting transmon - and by controlling their three energy levels in a digital manner, we experimentally simulate the parity-time symmetry-breaking phase transition. Our results indicate the potential advantage of multi-level (qudit) processors in simulating physical effects, where additional accessible levels can play the role of a controlled environment.Comment: 14 pages, 9 figure

Electromagnetic sources beyond common multipoles

The complete dynamic multipole expansion of electromagnetic sources contains more types of multipole terms than is conventionally perceived. The toroidal multipoles are one of the examples of such contributions that have been widely studied in recent years. Here we inspect more closely the other type of commonly overlooked terms known as the mean-square radii. In particular, we discuss both quantitative and qualitative aspects of the mean-square radii and provide a general geometrical framework for their visualization. We also consider the role of the mean-square radii in expanding the family of nontrivial nonradiating electromagnetic sources

Efficient variational synthesis of quantum circuits with coherent multi-start optimization

We consider the problem of the variational quantum circuit synthesis into a gate set consisting of the CNOT gate and arbitrary single-qubit (1q) gates with the primary target being the minimization of the CNOT count. First we note that along with the discrete architecture search suffering from the combinatorial explosion of complexity, optimization over 1q gates can also be a crucial roadblock due to the omnipresence of local minimums (well known in the context of variational quantum algorithms but apparently underappreciated in the context of the variational compiling). Taking the issue seriously, we make an extensive search over the initial conditions an essential part of our approach. Another key idea we propose is to use parametrized two-qubit (2q) controlled phase gates, which can interpolate between the identity gate and the CNOT gate, and allow a continuous relaxation of the discrete architecture search, which can be executed jointly with the optimization over 1q gates. This coherent optimization of the architecture together with 1q gates appears to work surprisingly well in practice, sometimes even outperforming optimization over 1q gates alone (for fixed optimal architectures). As illustrative examples and applications we derive 8 CNOT and T depth 3 decomposition of the 3q Toffoli gate on the nearest-neighbor topology, rediscover known best decompositions of the 4q Toffoli gate on all 4q topologies including a 1 CNOT gate improvement on the star-shaped topology, and propose decomposition of the 5q Toffoli gate on the nearest-neighbor topology with 48 CNOT gates. We also benchmark the performance of our approach on a number of 5q quantum circuits from the ibm_qx_mapping database showing that it is highly competitive with the existing software. The algorithm developed in this work is available as a Python package CPFlow.Comment: 12+4 pages, code available at https://github.com/idnm/cpflow, comments are welcome

Nonradiating anapole condition derived from Devaney-Wolf theorem and excited in a broken-symmetry dielectric particle

In this work, we first derive the nonradiating anapole condition with a straightforward theoretical demonstration exploiting one of the Devaney-Wolf theorems for nonradiating currents. Based on the equivalent volumetric and surface electromagnetic sources, it is possible to establish a unique compact conditions directly from Maxwell's Equations in order to ensure nonradiating anapole state. In addition, we support our theoretical findings with a numerical investigation on a broken-symmetry dielectric particle, building block of a metamaterial structure, demonstrating through a detailed multiple expansion the nonradiating anapole condition behind these peculiar destructive interactions. (C) 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreemen